denseDiscriminant -- dense discriminant (classical discriminant)

Synopsis

• Usage:

  denseDiscriminant(d,n)

  denseDiscriminant f

• Inputs:
  • (d,n), a pair of two integers representing, respectively, the degree and the number of variables of a Laurent polynomial $f$.
  • f, a ring element, a Laurent polynomial of degree $d$ in $n$ variables.
• Optional inputs:
  • Algorithm => ..., default value "Poisson"
  • CoefficientRing => ..., default value ZZ
• Outputs:
  • for (d,n), this is the same as sparseDiscriminant exponentsMatrix "generic polynomial of degree d in n variables";
  • for f, this is the same as affineDiscriminant(f).

Description

i1 : (d,n) := (2,3);

i2 : time Disc = denseDiscriminant(d,n)
 -- used 0.553202s (cpu); 0.422255s (thread); 0s (gc)

o2 = Disc

o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 2 |)
                                                           | 0 0 0 1 1 2 0 0 1 0 |
                                                           | 0 1 2 0 1 0 0 1 0 0 |

i3 : f = first genericLaurentPolynomials prepend(d,n:0)

        2               2                        2
o3 = a x  + a x x  + a x  + a x x  + a x x  + a x  + a x  + a x  + a x  + a
      9 1    8 1 2    5 2    7 1 3    4 2 3    2 3    6 1    3 2    1 3    0

o3 : ZZ[a ..a , b , c , d ][x ..x ]
         0   9   0   0   0   1   3

i4 : assert(Disc(f) == denseDiscriminant(f))

See also

• sparseDiscriminant -- sparse discriminant (A-discriminant)
• affineDiscriminant -- affine discriminant
• denseResultant -- dense resultant (classical resultant)
• exponentsMatrix -- exponents in one or more polynomials
• genericLaurentPolynomials -- generic (Laurent) polynomials